Cipollini look like small onions but actually are the bittersweet bulbs of the grape hyacinth. Fresh cipollini are available mostly in late summer and fall. If you can't find them, pearl onions work just as well.
In a small saucepan bring vinegar to boiling; reduce heat. Simmer, uncovered, for 5 minutes or until vinegar is reduced to 3 tablespoons. Set aside to cool.
Drain artichoke hearts, reserving 2 tablespoons liquid; set aside. On 4 long metal skewers alternately thread peppers, cipollini, and mushrooms, leaving a 1/4-inch space between pieces. In a small bowl combine reduced vinegar and reserved artichoke liquid. Brush half of the vinegar mixture over vegetables.
Grill skewers on the rack of an uncovered grill directly over medium heat for 8 to 10 minutes or until vegetables are tender, turning once and brushing often with remaining vinegar mixture. Remove vegetables from skewers and transfer to a large bowl; add drained artichokes, cheese, and basil. Season with salt and pepper to taste. Toss gently to combine. Makes 6 servings.